#9
The Simplex method
Exercise
Junk Jewelry Inc. makes both necklaces and bracelets from gold and platinum. The store has defined the following linear programming model for determining the number of necklaces and bracelets it needs to make in order to maximize profit.
Solve this linear programming model using the simplex method.
X_{1} = number of necklaces
X_{2} = number of bracelets
Z_{max} = 300X_{1} + 400X_{2 }(profit, $)
Subject to:
3X_{1} + 2X_{2} <=18 (gold,oz)
2X_{1} + 4X_{2} <= 20 (platinum, oz)
X_{2} <= 4 (demand, bracelets)
X_{1},X_{2} >= 0
Transform the constraints into equations.
Simplex Tableau for this Model
1^{st} Iteration
(Reference: Handout #8A)


C_{j} 
Basic Variables 
Quantity (RHS) 













Z_{j} 

C_{j}  Z_{j} 
1) Enter the model variables and the basic variables for the basic feasible solution. Enter the quantities of the basic variables.
2) Record the Cj values in the 1^{st} row and the 1^{st} column.
3) Record the coefficients for the decision variables and the slack variables in the constraint equations.
4) Compute and record the Zj values.
5) Compute and record the Cj – Zj values.
6) Is this the optimal solution? If not, determine the pivot column (the variable that will enter the solution). Determine the pivot row (the variable that will leave the solution). Note the pivot number.
Simplex Tableau for this Model
2nd Iteration
(Reference: Handout #8B)


C_{j} 
Basic Variables 
Quantity (RHS) 













Z_{j} 

C_{j}  Z_{j} 
1) Record the new basic variables and the Cj values for these variables. As you did in the first iteration, record all the decision variables and their coefficients from the objective function.
2) Compute and record the new pivot row values.
3) Compute and record the remaining rows (in this case, there are two).
4) Compute and record the Zj row.
5) Compute and record the Cj – Zj row.
6) Is this the optimal solution? If not, determine the pivot column (the variable that will enter the solution). Determine the pivot row (the variable that will leave the solution). Note the pivot number.
Simplex Tableau for this Model
3rd Iteration


C_{j} 
Basic Variables 
Quantity (RHS) 













Z_{j} 

C_{j}  Z_{j} 
1) Record the new basic variables and the Cj values for these variables. As you did in the first iteration, record all the decision variables and their coefficients from the objective function.
2) Compute and record the new pivot row values.
3) Compute and record the remaining rows (in this case, there are two).
4) Compute and record the Zj row.
5) Compute and record the Cj – Zj row.
6) Is this the optimal solution? If not, determine the pivot column (the variable that will enter the solution). Determine the pivot row (the variable that will leave the solution). Note the pivot number.
Simplex Tableau for this Model
4th Iteration


C_{j} 
Basic Variables 
Quantity (RHS) 













Z_{j} 

C_{j}  Z_{j} 
1) Record the new basic variables and the Cj values for these variables. As you did in the first iteration, record all the decision variables and their coefficients from the objective function.
2) Compute and record the new pivot row values.
3) Compute and record the remaining rows (in this case, there are two).
4) Compute and record the Zj row.
5) Compute and record the Cj – Zj row.
6) Is this the optimal solution?